A → differential equation composed of
→ continuous
→ differentiable functions for which certain conditions
are fulfilled. The equation M(x,y)dx + N(x,y)dy = 0 is called exact if
M(x,y) and N(x,y) are continuous differentiable functions
for which the following relationship is fulfilled:
∂M/∂y = ∂N/∂x, and ∂M/∂y
and ∂N/∂x are continuous in some region.

An equation in which the → dependent variabley
and all its differential coefficients occur only
in the first degree. A linear differential equation of → order
order n has the form: fn(x)y(n) + fn-1(x)y(n-1) + ... +
f1(x)y' + f0(x)y = Q(x),
where f0(x), f1(x), ..., fn(x)
and Q(x) are each continuous functions of x defined on a common
interval I and fn(x)≠ 0 in I. A linear differential
equation cannot have, for example, terms such as y2 or
(y')1/2. See also:
→ homogeneous linear differential equation;
→ nonhomogeneous linear differential equation.